We consider the competition among quantity setting players in a deterministic nonlinear oligopoly framework characterized by an isoelastic demand curve. Players are characterized by having heterogeneous decisional mechanisms to set their outputs: some players are imitators, while the remaining others adopt a rational-like rule according to which their past decisions are adjusted towards their static expectation best response. The Cournot-Nash production level is a stationary state of our model together with a further production level that can be interpreted as the competitive outcome in case only imitators are present. We found that both the number of players and the relative fraction of imitators influence stability of the Cournot-Nash equilibrium with an ambiguous role, and double instability thresholds may be observed. Global analysis shows that a wide variety of complex dynamic scenarios emerge. Chaotic trajectories as well as multi-stabilities, where different attractors coexist, are robust phenomena that can be observed for a wide spectrum of parameter sets.
References
1.
H. N.
Agiza
and A. A.
Elsadany
, “Nonlinear dynamics in the cournot duopoly game with heterogeneous players
,” Physica A
320
, 512
–524
(2003
).2.
H. N.
Agiza
and A. A.
Elsadany
, “Chaotic dynamics in nonlinear duopoly game with heterogeneous players
,” Appl. Math. Comput.
149
(3
), 843
–860
(2004
).3.
H. N.
Agiza
, A. S.
Omar Hegazi
, and A. A.
Elsadany
, “Complex dynamics and synchronization of a duopoly game with bounded rationality
,” Math. Comput. Simul.
58
(2
), 133
–146
(2002
).4.
A.
Agliari
, G. I.
Bischi
, R.
Dieci
, and L.
Gardini
, “Global bifurcations of closed invariant curves in two-dimensional maps: A computer assisted study
,” Int. J. Bifurcation Chaos
15
(04
), 1285
–1328
(2005
).5.
J.
Andaluz
, A. A.
Elsadany
, and G.
Jarne
, “Nonlinear cournot and bertrand-type dynamic triopoly with differentiated products and heterogeneous expectations
,” Math. Comput. Simul.
132
, 86
–99
(2017
).6.
J.
Andaluz
and G.
Jarne
, “On the dynamics of economic games based on product differentiation
,” Math. Comput. Simul.
113
, 16
–27
(2015
).7.
N.
Angelini
, R.
Dieci
, and F.
Nardini
, “Bifurcation analysis of a dynamic duopoly model with heterogeneous costs and behavioural rules
,” Math. Comput. Simul.
79
(10
), 3179
–3196
(2009
).8.
J.
Apesteguia
, S.
Huck
, and J.
Oechssler
, “Imitation - theory and experimental evidence
,” J. Econ. Theory
136
(1
), 217
–235
(2007
).9.
J.
Apesteguia
, S.
Huck
, J.
Oechssler
, and S.
Weidenholzer
, “Imitation and the evolution of walrasian behavior: Theoretically fragile but behaviorally robust
,” J. Econ. Theory
145
(5
), 1603
–1617
(2010
).10.
S. S.
Askar
, “Complex dynamic properties of cournot duopoly games with convex and log-concave demand function
,” Oper. Res. Lett.
42
(1
), 85
–90
(2014
).11.
S. S.
Askar
, “The rise of complex phenomena in cournot duopoly games due to demand functions without inflection points
,” Commun. Nonlinear Sci. Numer. Simul.
19
(6
), 1918
–1925
(2014
).12.
M.
Bigoni
and M.
Fort
, “Information and learning in oligopoly: An experiment
,” Games Econ. Behav.
81
, 192
–214
(2013
).13.
G. I.
Bischi
, M.
Gallegati
, and A. K.
Naimzada
, “Symmetry-breaking bifurcations and representative firm in dynamic duopoly games
,” Ann. Oper. Res.
89
, 252
–271
(1999
).14.
G. I.
Bischi
, M.
Kopel
, and A. K.
Naimzada
, “On a rent-seeking game described by a non-invertible iterated map with denominator
,” Nonlinear Anal.: Theory, Methods Appl.
47
(8
), 5309
–5324
(2001
).15.
G. I.
Bischi
and A. K.
Naimzada
, “Global analysis of a dynamic duopoly game with bounded rationality
,” in Advances in Dynamic Games and Applications
(Springer
, 2000
), pp. 361
–385
.16.
G. I.
Bischi
, A. K.
Naimzada
, and L.
Sbragia
, “Oligopoly games with local monopolistic approximation
,” J. Econ. Behav. Org.
62
(3
), 371
–388
(2007
).17.
F.
Cavalli
and A.
Naimzada
, “Nonlinear dynamics and convergence speed of heterogeneous cournot duopolies involving best response mechanisms with different degrees of rationality
,” Nonlinear Dyn.
81
(1-2
), 967
–979
(2015
).18.
F.
Cavalli
, A.
Naimzada
, and M.
Pireddu
, “Heterogeneity and the (de) stabilizing role of rationality
,” Chaos, Solitons Fractals
79
, 226
–244
(2015
).19.
F.
Cavalli
and A. K.
Naimzada
, “A Cournot duopoly game with heterogeneous players: Nonlinear dynamics of the gradient rule versus local monopolistic approach
,” Appl. Math. Comput.
249
, 382
–388
(2014
).20.
F.
Cavalli
, A. K.
Naimzada
, and F.
Tramontana
, “Nonlinear dynamics and global analysis of a heterogeneous cournot duopoly with a local monopolistic approach versus a gradient rule with endogenous reactivity
,” Commun. Nonlinear Sci. Numer. Simul.
23
(1
), 245
–262
(2015
).21.
L. C.
Baiardi
and A. K.
Naimzada
, “Experimental oligopolies modeling: A dynamic approach based on heterogeneous behaviors
,” Commun. Nonlinear Sci. Numer. Simul.
58
, 47
–62
(2017
).22.
W. J.
Den Haan
, “The importance of the number of different agents in a heterogeneous asset-pricing model
,” J. Econ. Dyn. Control
25
(5
), 721
–746
(2001
).23.
A.
Dohtani
, T.
Inaba
, and H.
Osaka
, “Corridor stability of the neoclassical steady state
,” in Time and Space in Economics
(Springer
, 2007
), pp. 129
–143
.24.
L.
Fanti
, L.
Gori
, and M.
Sodini
, “Nonlinear dynamics in a Cournot duopoly with isoelastic demand
,” Math. Comput. Simul.
108
, 129
–143
(2015
).25.
S.
Huck
, H.-T.
Normann
, and J.
Oechssler
, “Two are few and four are many: Number effects in experimental oligopolies
,” J. Econ. Behav. Org.
53
(4
), 435
–446
(2004
).26.
A.
Leijonhufvud
, “Effective demand failures
,” Swed. J. Econ.
75
, 27
–48
(1973
).27.
D.
Leonard
and K.
Nishimura
, “Nonlinear dynamics in the Cournot model without full information
,” Ann. Oper. Res.
89
, 165
–173
(1999
).28.
A.
Matsumoto
and F.
Szidarovszky
, “Stability, bifurcation, and chaos in n-firm nonlinear cournot games
,” Discrete Dyn. Nat. Soc.
2011
, 1
–22
.29.
A. K.
Naimzada
and G.
Ricchiuti
, “Monopoly with local knowledge of demand function
,” Econ. Modell.
28
(1
), 299
–307
(2011
).30.
A. K.
Naimzada
and L.
Sbragia
, “Oligopoly games with nonlinear demand and cost functions: Two boundedly rational adjustment processes
,” Chaos, Solitons Fractals
29
(3
), 707
–722
(2006
).31.
A. K.
Naimzada
and F.
Tramontana
, “Controlling chaos through local knowledge
,” Chaos, Solitons Fractals
42
(4
), 2439
–2449
(2009
).32.
A. K.
Naimzada
and F.
Tramontana
, “Two different routes to complex dynamics in an heterogeneous triopoly game
,” J. Differ. Equations Appl.
21
(7
), 553
–563
(2015
).33.
J.
Oechssler
, A.
Roomets
, and S.
Roth
, “From imitation to collusion: A replication
,” J. Econ. Sci. Assoc.
2
, 13
–21
(2016
).34.
T.
Offerman
, J.
Potters
, and J.
Sonnemans
, “Imitation and belief learning in an oligopoly experiment
,” Rev. Econ. Stud.
69
(4
), 973
–997
(2002
).35.
M.
Pireddu
, “Chaotic dynamics in three dimensions: A topological proof for a triopoly game model
,” Nonlinear Anal.: Real World Appl.
25
, 79
–95
(2015
).36.
T.
Puu
, “Chaos in duopoly pricing
,” Chaos, Solitons Fractals
1
(6
), 573
–581
(1991
).37.
K.
Schlag
, “Why imitate, and if so, how?: A boundedly rational approach to multi-armed bandits
,” J. Econ. Theory
78
(1
), 130
–156
(1998
).38.
R.
Theocharis
, “On the stability of the Cournot solution on the oligopoly problem
,” Rev. Econ. Stud.
27
(2
), 133
–134
(1960
).39.
F.
Tramontana
, “Heterogeneous duopoly with isoelastic demand function
,” Econ. Modell.
27
(1
), 350
–357
(2010
).40.
F.
Tramontana
, A. A.
Elsadany
, B.
Xin
, and H. N.
Agiza
, “Local stability of the Cournot solution with increasing heterogeneous competitors
,” Nonlinear Anal.: Real World Appl.
26
, 150
–160
(2015
).41.
J.
Tuinstra
, “A price adjustment process in a model of monopolistic competition
,” Int. Game Theory Rev.
6
(03
), 417
–442
(2004
).42.
F.
Vega-Redondo
, “The evolution of walrasian behavior
,” Econometrica
65
(2
), 375
–384
(1997
).© 2018 Author(s).
2018
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